Chapter 23 Small gaps between primes (proofs)
\(k,\ell\text{.}\)\(x\)\(\infty\text{.}\)\(\calH = (h_1, \dots, h_k)\)\(k\)\(1, \dots, H\text{,}\)\(H \leq \lambda \log x\)\(\lambda\text{.}\)\(p\)\(v_{\calH}(p) = \#\Omega(p)\text{.}\)\(d\text{.}\)\(R \leq x^{1/2}/(\log x)^C\text{,}\)\(C\)\(k,\ell\)\(c>0\)\(k, \ell\text{,}\)\(C\)\(k,\ell\)
\begin{equation*}
\sum_{x \lt n\leq 2x} a(n) = \frac{\frakS(\calH) (k+\ell)!^2}{(k+2\ell)! (\log R)^{2k+2\ell}}
\binom{2\ell}{\ell} x (\log R)^{k+2\ell}
+ O\left( \frac{x (\log x)^{k+2\ell-1} (\log \log x)^c}{(\log R)^{2k+2\ell} }
\right).
\end{equation*}
\(a(n)\text{,}\)\(d_1, d_2\)\(\rho(d_1) \rho(d_2)\)\(x \lt n \leq 2x\)\(n \in \Omega(d_1), \Omega(d_2)\text{.}\)\(|\Omega(d)| \leq \tau_k(d)\text{,}\)\(O(R^2 (\log R)^c)\text{.}\)\((\alpha)\)\(\alpha - i\infty
\to \alpha + i \infty\text{.}\)\(v_{\calH}(p) = k\)\(p\text{,}\)\(\Real(s_1), \Real(s_2) > -c\text{.}\)\(\min\{\Real(s_1), \Real(s_2), 0\} = \sigma
\geq -c\text{,}\)\(p \leq k^2\)\(p > H\text{;}\)\(k^2 \lt p \leq H\text{.}\)\(U = \exp(\sqrt{\log x})\text{.}\)\(s_1\)\(L_1 = (\log U)^{-1} + it\text{,}\)\(s_2\)\(L_2 = (2 \log U)^{-1} + it\text{.}\)\(|t| \leq U\)\(|t| \leq U/2\text{,}\)\(s_1\)\(L'_1 = -(\log U)^{-1}
+ it\)\(|t| \leq U\text{;}\)\(s_1=0\)\(s_1=-s_2\text{.}\)\(s_1 = -s_2\)\(|s_1+s_2| = (\log x)^{-1}\text{.}\)\(G(s_1,s_2,\Omega) = O((\log \log x)^c)\text{,}\)\(R^{s_1+s_2} = O(1)\text{,}\)\(\zeta(s_1+s_2+1) = O(\log x)\text{.}\)\(\Res_{s_1 = 0}\text{;}\)\(\ell+1\text{.}\)\(Z(s_1, s_2, \calH)\)\((0,0)\text{,}\)\(s_2\)\(L_2': -(2 \log U)^{-1} + it\)\(|t| \leq U/2\text{.}\)\(O(\exp(-c \sqrt{\log x}))\text{,}\)\(s_2 = 0\text{.}\)\((0,0)\text{,}\)\(\rho > 0\)\(C_1\)\(|s_1| = \rho\text{,}\)\(C_2\)\(|s_2| = 2\rho\text{.}\)\(s, \xi\)\(s_1 = s\)\(s_2 = s\xi\text{,}\)\(C: |s| = \rho\)\(C': |\xi| = 2\text{.}\)\(s\)\(\binom{2\ell}{\ell}\text{,}\)\(h \in \calH\)\(n+h\)\(a(n, \calH) = a(n, \calH,h)\text{.}\)\(c>0\)\(k, \ell\text{,}\)\(C\)\(k,\ell\)\(h \notin \calH\text{,}\)\(g\)\(g(p) = v_{\calH}(p) - 1\text{,}\)\(h \in \calH\text{,}\)\(\log x\text{,}\)\(\log x\)\(\Lambda(n)\text{;}\)\(v_{\calH,h}(p) = p\)\(p\text{.}\)\(s_1 = 0\)\(s_2 = 0\text{,}\)\(v_{\calH,h}(p) = 0\text{.}\)\(p \leq k + 1\text{,}\)\(\frakS(\calH \cup \{h\}) = 0\text{.}\)
